Strong Nash equilibrium (SNE) is an appealing solution concept when rational agents can form coalitions. A strategy profile is an SNE if no coalition of agents can benefit by deviating. We present the first general-purpose algorithms for SNE finding in games with more than two agents. An SNE must simultaneously be a Nash equilibrium (NE) and the optimal solution of multiple non-convex optimization problems. This makes even the derivation of necessary and sufficient mathematical equilibrium constraints difficult. We show that forcing an SNE to be resilient only to pure-strategy deviations by coalitions, unlike for NEs, is only a necessary condition here. Second, we show that the application of Karush-Kuhn-Tucker conditions leads to another set of necessary conditions that are not sufficient. Third, we show that forcing the Pareto efficiency of an SNE for each coalition with respect to coalition correlated strategies is sufficient but not necessary. We then develop a tree search algorithm for SNE finding. At each node, it calls an oracle to suggest a candidate SNE and then verifies the candidate. We show that our new necessary conditions can be leveraged to make the oracle more powerful. Experiments validate the overall approach and show that the new conditions significantly reduce search tree size compared to using NE conditions alone.

In this paper, a general Maximum K-Min approach for classification is proposed. With the physical meaning of optimizing the classification confidence of the K worst instances, Maximum K-Min Gain/Minimum K-Max Loss (MKM) criterion is introduced. To make the original optimization problem with combinational constraints computationally tractable, the optimization techniques are adopted and a general compact representation lemma for MKM Criterion is summarized. Based on the lemma, a Nonlinear Maximum K-Min (NMKM) classifier and a Semi-supervised Maximum K-Min (SMKM) classifier are presented for traditional classification task and semi-supervised classification task respectively. Based on the experiment results of publicly available datasets, our Maximum K-Min methods have achieved competitive performance when comparing against Hinge Loss classifiers.

Stochastic local search (SLS) algorithms are well known for their ability to efficiently find models of random instances of the Boolean satisfiablity (SAT) problem. One of the most famous SLS algorithms for SAT is WalkSAT, which is an initial algorithm that has wide influence among modern SLS algorithms. Recently, there has been increasing interest in WalkSAT, due to the discovery of its great power on large random 3-SAT instances. However, the performance of WalkSAT on random $k$-SAT instances with $k>3$ lags far behind. Indeed, there have been few works in improving SLS algorithms for such instances. This work takes a large step towards this direction. We propose a novel concept namely $multilevel$ $make$. Based on this concept, we design a scoring function called $linear$ $make$, which is utilized to break ties in WalkSAT, leading to a new algorithm called WalkSAT$lm$. Our experimental results on random 5-SAT and 7-SAT instances show that WalkSAT$lm$ improves WalkSAT by orders of magnitudes. Moreover, WalkSAT$lm$ significantly outperforms state-of-the-art SLS solvers on random 5-SAT instances, while competes well on random 7-SAT ones. Additionally, WalkSAT$lm$ performs very well on random instances from SAT Challenge 2012, indicating its robustness.

It is known that an equilibrium of an infinitely repeated two-player game (with limit average payoffs) can be computed in polynomial time, as follows: according to the folk theorem, we compute minimax strategies for both players to calculate the punishment values, and subsequently find a mixture over outcomes that exceeds these punishment values. However, for very large games, even computing minimax strategies can be prohibitive. In this paper, we propose an algorithmic framework for computing equilibria of repeated games that does not require linear programming and that does not necessarily need to inspect all payoffs of the game. This algorithm necessarily sometimes fails to compute an equilibrium, but we mathematically demonstrate that most of the time it succeeds quickly on uniformly random games, and experimentally demonstrate this for other classes of games. This also holds for games with more than two players, for which no efficient general algorithms are known.

We present a dynamic algorithm for solving the Longest Common Subsequence Problem using Ant Colony Optimization Technique. The Ant Colony Optimization Technique has been applied to solve many problems in Optimization Theory, Machine Learning and Telecommunication Networks etc. In particular, application of this theory in NP-Hard Problems has a remarkable significance. Given two strings, the traditional technique for finding Longest Common Subsequence is based on Dynamic Programming which consists of creating a recurrence relation and filling a table of size . The proposed algorithm draws analogy with behavior of ant colonies function and this new computational paradigm is known as Ant System. It is a viable new approach to Stochastic Combinatorial Optimization. The main characteristics of this model are positive feedback, distributed computation, and the use of constructive greedy heuristic. Positive feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence and greedy heuristic helps find acceptable solutions in minimum number of stages. We apply the proposed methodology to Longest Common Subsequence Problem and give the simulation results. The effectiveness of this approach is demonstrated by efficient Computational Complexity. To the best of our knowledge, this is the first Ant Colony Optimization Algorithm for Longest Common Subsequence Problem.

ETP is NP Hard combinatorial optimization problem. It has received tremendous research attention during the past few years given its wide use in universities. In this Paper, we develop three mathematical models for NSOU, Kolkata, India using FILP technique. To deal with impreciseness and vagueness we model various allocation variables through fuzzy numbers. The solution to the problem is obtained using Fuzzy number ranking method. Each feasible solution has fuzzy number obtained by Fuzzy objective function. The different FILP technique performance are demonstrated by experimental data generated through extensive simulation from NSOU, Kolkata, India in terms of its execution times. The proposed FILP models are compared with commonly used heuristic viz. ILP approach on experimental data which gives an idea about quality of heuristic. The techniques are also compared with different Artificial Intelligence based heuristics for ETP with respect to best and mean cost as well as execution time measures on Carter benchmark datasets to illustrate its effectiveness. FILP takes an appreciable amount of time to generate satisfactory solution in comparison to other heuristics. The formulation thus serves as good benchmark for other heuristics. The experimental study presented here focuses on producing a methodology that generalizes well over spectrum of techniques that generates significant results for one or more datasets. The performance of FILP model is finally compared to the best results cited in literature for Carter benchmarks to assess its potential. The problem can be further reduced by formulating with lesser number of allocation variables it without affecting optimality of solution obtained. FLIP model for ETP can also be adapted to solve other ETP as well as combinatorial optimization problems.

Transportation Problem is an important problem which has been widely studied in Operations Research domain. It has been often used to simulate different real life problems. In particular, application of this Problem in NP Hard Problems has a remarkable significance. In this Paper, we present the closed, bounded and non empty feasible region of the transportation problem using fuzzy trapezoidal numbers which ensures the existence of an optimal solution to the balanced transportation problem. The multivalued nature of Fuzzy Sets allows handling of uncertainty and vagueness involved in the cost values of each cells in the transportation table. For finding the initial solution of the transportation problem we use the Fuzzy Vogel Approximation Method and for determining the optimality of the obtained solution Fuzzy Modified Distribution Method is used. The fuzzification of the cost of the transportation problem is discussed with the help of a numerical example. Finally, we discuss the computational complexity involved in the problem. To the best of our knowledge, this is the first work on obtaining the solution of the transportation problem using fuzzy trapezoidal numbers.

Transportation Problem is an important aspect which has been widely studied in Operations Research domain. It has been studied to simulate different real life problems. In particular, application of this Problem in NP- Hard Problems has a remarkable significance. In this Paper, we present a comparative study of Transportation Problem through Probabilistic and Fuzzy Uncertainties. Fuzzy Logic is a computational paradigm that generalizes classical two-valued logic for reasoning under uncertainty. In order to achieve this, the notation of membership in a set needs to become a matter of degree. By doing this we accomplish two things viz., (i) ease of describing human knowledge involving vague concepts and (ii) enhanced ability to develop cost-effective solution to real-world problem. The multi-valued nature of Fuzzy Sets allows handling uncertain and vague information. It is a model-less approach and a clever disguise of Probability Theory. We give comparative simulation results of both approaches and discuss the Computational Complexity. To the best of our knowledge, this is the first work on comparative study of Transportation Problem using Probabilistic and Fuzzy Uncertainties.

Pattern databases have been successfully applied to several problems. Their use assumes that the goal state is known, and once the pattern database is built, commonly it can be used by all instances. However, in Sokoban, before solving the puzzle, the goal position of each stone is unknown. Moreover, each Sokoban instance has its own state space search. In this paper we apply pattern databases to Sokoban. The proposed approach uses an instance decomposition, that allows multiple possible goal states to be abstracted into a single state. Thus, an instance dependent pattern database is employed. Experiments with the standard set of instances show that the proposed approach overcomes the current best lower bounds in initial states for several instances. Furthermore, three of these new best lower bounds match exactly with the optimal solution length. Finally, we run experiments of 5 million explored states for each instance. Nine instances were solved with optimality guarantees, while only four instances were solved under the same conditions by previous methods.

Boosting methods are highly popular and effective supervised learning methods which combine weak learners into a single accurate model with good statistical performance. In this paper, we analyze two well-known boosting methods, AdaBoost and Incremental Forward Stagewise Regression (FS$_\varepsilon$), by establishing their precise connections to the Mirror Descent algorithm, which is a first-order method in convex optimization. As a consequence of these connections we obtain novel computational guarantees for these boosting methods. In particular, we characterize convergence bounds of AdaBoost, related to both the margin and log-exponential loss function, for any step-size sequence. Furthermore, this paper presents, for the first time, precise computational complexity results for FS$_\varepsilon$.